A Fourier Analytic Criterion for Decoding on the BSC

We present an approach to showing that a linear code is resilient to random errors. We then use this approach to obtain decoding results for both transitive codes and Reed-Muller codes. We give three kinds of results about linear codes in general, and transitive linear codes in particular. 1) We give a tight bound on the weight distribution of every transitive linear code $C \subseteq \mathbb{F}_2^N$: $\Pr_{c \in C}[|c| = \alpha N] \leq 2^{-(1-h(\alpha)) \mathsf{dim}(C)}$. 2) We give a Fourier analytic criterion that certifies that a linear code $C$ can be decoded on the binary symmetric channel. Let $L_w(x)$ denote the level function that is $1$ if $|x| =w$ and $0$ otherwise, and let $C^\perp$ denote the dual code of $C$. We show that bounds on $\mathbb{E}_{c \in C^{\perp}}[ \hat{L}_{\epsilon N}(c)^2]$ imply that $C$ recovers from errors on the binary symmetric channel with parameter $\epsilon$. Weaker bounds can be used to obtain list-decoding results using similar methods. 3) Motivated by the above results, we use complex analysis to give tight estimates for the Fourier coefficients of the level function. We then combine these estimates with our weight distribution bound to give list-decoding results for transitive linear codes and Reed-Muller codes.